As a consequence of Lemma REF , there exists a subsequence of \(\lbrace \Omega _n\rbrace \) , which shall be denoted in the same manner, and a domain \(\Omega ^*\)  such that \(\Omega _n\rightarrow \Omega ^*\)  in \(\operatorname{\mathcal {O}_{ad}}\, \) .
By definition, \(\mathcal {J}^*\le \mathcal {J}(\Omega ^*)\) . However, since \(-J_1\) , \(-J_2\) , and \(P\)  (for \(P\) , see Proposition 2.3.7 in [1]}) are lower semicontinuous,
\(\mathcal {J}(\Omega ^*) & = P(\Omega ^*) - (J_1(\Omega ^*) + J_2(\Omega ^*)) \le \liminf _{n\rightarrow \infty } [P(\Omega _n) - (J_1(\Omega _n) + J_2(\Omega _n))]\\ & = \liminf _{n\rightarrow \infty } \mathcal {J}(\Omega _n) = \mathcal {J}^*.\) 
